Algebra Know-It-ALL

266 Two-by-Two Linear Graphs

precisely draw the line through two points that are close together, we can find a third point farther out on

that line and use it. For example, in the equation 8x+ 4 y= 16 for the line through (0, 4) and (5, −6), we

can plug in x= 30 and easily solve for y using the SI form:

y=− 2 x+ 4

=− 2 × 30 + 4

=− 60 + 4

=− 56

That tells us that the point (30, −56) is on the line. It is shown as an open circle in Fig. 17-1. This point

is far enough away from (0, 4) so we can easily draw the line.

Here’s a challenge!

Revisit the airplane challenge from Chap. 16. While flying straight into the wind, an airplane has a

groundspeed of 750 km/h. When flying straight downwind at the same airspeed, the plane has a ground-

speed of 990 km/h. Let x represent the airspeed of the plane, and let y represent the speed of the wind,

both in kilometers per hour. Then

x−y= 750

and

x+y= 990

Graph these two equations, showing that the airspeed of the plane is 870 km/h, and the wind is blowing

at 120 km/h.

Solution

Let’s write down the SI forms of the equations that we derived before we mixed them. They are

y=x− 750

and

y=−x+ 990

They-intercepts are −750 and 990, so we can plot the points (0, −750) and (0, 990) on a Cartesian

plane where x is the independent variable and y is the dependent variable. The solution we obtained

in Chap. 16 was x= 870 and y= 120, giving us another point with the ordered pair (870,120).

Figure 17-2 shows these three points and the lines through them. That’s how this two-by-two linear

system looks when graphed. The reference points we found are well separated, so the lines are easy

to draw.